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Theorem elpcliN 33537
Description: Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
elpcli.s  |-  S  =  ( PSubSp `  K )
elpcli.c  |-  U  =  ( PCl `  K
)
Assertion
Ref Expression
elpcliN  |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  /\  Q  e.  ( U `  X )
)  ->  Q  e.  Y )

Proof of Theorem elpcliN
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simp1 988 . . . . . 6  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  K  e.  V )
2 simp2 989 . . . . . . 7  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  X  C_  Y )
3 eqid 2443 . . . . . . . . 9  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 elpcli.s . . . . . . . . 9  |-  S  =  ( PSubSp `  K )
53, 4psubssat 33398 . . . . . . . 8  |-  ( ( K  e.  V  /\  Y  e.  S )  ->  Y  C_  ( Atoms `  K ) )
653adant2 1007 . . . . . . 7  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  Y  C_  ( Atoms `  K )
)
72, 6sstrd 3366 . . . . . 6  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  X  C_  ( Atoms `  K )
)
8 elpcli.c . . . . . . 7  |-  U  =  ( PCl `  K
)
93, 4, 8pclvalN 33534 . . . . . 6  |-  ( ( K  e.  V  /\  X  C_  ( Atoms `  K
) )  ->  ( U `  X )  =  |^| { z  e.  S  |  X  C_  z } )
101, 7, 9syl2anc 661 . . . . 5  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( U `  X )  =  |^| { z  e.  S  |  X  C_  z } )
1110eleq2d 2510 . . . 4  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( Q  e.  ( U `  X )  <->  Q  e.  |^|
{ z  e.  S  |  X  C_  z } ) )
12 elintrabg 4141 . . . . 5  |-  ( Q  e.  |^| { z  e.  S  |  X  C_  z }  ->  ( Q  e.  |^| { z  e.  S  |  X  C_  z }  <->  A. z  e.  S  ( X  C_  z  ->  Q  e.  z )
) )
1312ibi 241 . . . 4  |-  ( Q  e.  |^| { z  e.  S  |  X  C_  z }  ->  A. z  e.  S  ( X  C_  z  ->  Q  e.  z ) )
1411, 13syl6bi 228 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( Q  e.  ( U `  X )  ->  A. z  e.  S  ( X  C_  z  ->  Q  e.  z ) ) )
15 sseq2 3378 . . . . . . . 8  |-  ( z  =  Y  ->  ( X  C_  z  <->  X  C_  Y
) )
16 eleq2 2504 . . . . . . . 8  |-  ( z  =  Y  ->  ( Q  e.  z  <->  Q  e.  Y ) )
1715, 16imbi12d 320 . . . . . . 7  |-  ( z  =  Y  ->  (
( X  C_  z  ->  Q  e.  z )  <-> 
( X  C_  Y  ->  Q  e.  Y ) ) )
1817rspccv 3070 . . . . . 6  |-  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  ( Y  e.  S  ->  ( X  C_  Y  ->  Q  e.  Y ) ) )
1918com13 80 . . . . 5  |-  ( X 
C_  Y  ->  ( Y  e.  S  ->  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  Q  e.  Y ) ) )
2019imp 429 . . . 4  |-  ( ( X  C_  Y  /\  Y  e.  S )  ->  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  Q  e.  Y ) )
21203adant1 1006 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  Q  e.  Y ) )
2214, 21syld 44 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( Q  e.  ( U `  X )  ->  Q  e.  Y ) )
2322imp 429 1  |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  /\  Q  e.  ( U `  X )
)  ->  Q  e.  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   {crab 2719    C_ wss 3328   |^|cint 4128   ` cfv 5418   Atomscatm 32908   PSubSpcpsubsp 33140   PClcpclN 33531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-psubsp 33147  df-pclN 33532
This theorem is referenced by:  pclfinclN  33594
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